= Solution
The antipodal action fixes the origin. Away from the origin it acts freely, so the quotient is locally Euclidean there. A small punctured neighborhood of the image of the origin, however, has link
$$
S^2/(x\sim-x)=\mathbb{RP}^2,
$$
the <real projective plane>, whereas the link of a point in a three-dimensional <smooth manifold> is $S^2$. Equivalently, the neighborhood is a cone on $\mathbb{RP}^2$ and cannot be a three-ball. Thus
$$
\boxed{\mathbb R^3/(x\sim-x)\text{ is not a manifold}.}
$$
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